Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $y \neq 0$. $p = \dfrac{-5}{8y + 64} \div \dfrac{7y}{5(y + 8)} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{-5}{8y + 64} \times \dfrac{5(y + 8)}{7y} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ -5 \times 5(y + 8) } { (8y + 64) \times 7y } $ $ p = \dfrac {-5 \times 5(y + 8)} {7y \times 8(y + 8)} $ $ p = \dfrac{-25(y + 8)}{56y(y + 8)} $ We can cancel the $y + 8$ so long as $y + 8 \neq 0$ Therefore $y \neq -8$ $p = \dfrac{-25 \cancel{(y + 8})}{56y \cancel{(y + 8)}} = -\dfrac{25}{56y} $